In recent years great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a variety of techniques from Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of problems connected with dispersive equations, such as the derivation of a certain nonlinear Schrodinger equations from a quantum many-particles system, periodic Strichartz estimates, the concept of energy transfer, the invariance of a Gibbs measure associated to an infinite dimension Hamiltonian system and non-squeezing theorems for such systems when they also enjoy a symplectic structure.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will discuss an inverse problem for the wave equation, where a collection (array) of sensors probes an unknown heterogeneous
medium with waves and measures the echoes.The goal is to determine scattering structures in the medium modeled by a reflectivity function. Much of the existing imaging methodology is based on a linear least squares data fit approach. However, the mapping between the reflectivity and the wave measured at the array is nonlinear and the resulting images have artifacts. I will show how to use a reduced order model (ROM) approach to solve the inverse scattering problem. The ROM is data driven i.e., it is constructed from the data, with no knowledge of the medium. It approximates the wave propagator, which is the operator that maps the wave from one time step to the next. I will show how to use the ROM to: (1) Remove the multiple scattering (nonlinear) effects from the data, which can then be used with any linearized inversion algorithm. (2) Obtain a well conditioned quantitative inversion algorithm for estimating the reflectivity.

Reaction-diffusion systems with strong interaction terms appear in many multi-species physical problems as well as in population dynamics. The qualitative properties of the solutions and their limiting profiles in different regimes have been at the center of the community's attention in recent years. A prototypical example is the system of equations
\[\left\{\begin{array}{l}
-\Delta u+a_1u = b_1|u|^{p+q-2}u+cp|u|^{p-2}|v|^qu,\\
-\Delta v+a_2v = b_2|v|^{p+q-2}v+cq|u|^{p}|v|^{q-2}v
\end{array}
\right.
\]
in a domain $\Omega\subset \mathbb{R}^N$ which appears, for example, when looking for solitary wave solutions for Bose-Einstein condensates of two different hyperfine states which overlap in space. The sign of $b_i$ reflects the interaction of the particles within each single state. If $b_i$ is positive, the self interaction is attractive (focusing problems). The sign of $c$, on the other hand, reflects the interaction of particles in different states. This interaction is attractive if $c>0$ and repulsive if $c<0$. If the condensates repel, they eventually separate spatially giving rise to a free boundary. Similar phenomena occurs for many species systems. As a model problem, we consider the system of stationary equations:
\[
\begin{cases}
-\Delta u_i=f_i(u_i)-\beta u_i\sum_{j\neq i}g_{ij}(u_j)\;\\
u_i>0\;.
\end{cases}
\]
The cases $g_{ij}(s)=\beta_{ij}s$ (Lotka-Volterra competitive interactions) and $g_{ij}(s)=\beta_{ij}s^2$ (gradient system for Gross-Pitaevskii energies) are of particular interest in the applications to population dynamics and theoretical physics respectively.
Phase separation and has been described in the recent literature, both physical and mathematical. Relevant connections have been established with optimal partition problems involving spectral functionals. The classification of entire solutions and the geometric aspects of phase separation are of fundamental importance as well. We intend to focus on the most recent developments of the theory in connection with problems featuring anomalous diffusions, non-local and non symmetric interactions.

The lecture focuses on the partial differential equations arising from geometric flows. We will first introduce the curve shortening flow and the mean curvature flow then highlight the similarities and differences with the heat equation. We will then discuss the various questions arising from the study of flows.

When does a given set contain a copy of your favourite pattern (for example, specially arranged points on a line or a spiral, or the
vertices of a polyhedron)? Does the answer depend on how thin the set is in some quantifiable sense? Problems involving identification of prescribed configurations under varying interpretations of size have been vigorously pursued both in the discrete and continuous setting, often with spectacular results that run contrary to intuition. Yet many deceptively simple questions remain open. I will survey the literature in this area, emphasizing some of the landmark results that focus on different aspects of the problem.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.